Answer:
a) The population of prairie dogs after nine months is 280.
b) P(t) = 30 + 30 · t - t²/4 for 0 ≤ t ≤ 60
Explanation:
Hi there!
We have the following information:
The initial population is P(0) = 30.
The rate of growth of the population is the following:
P´(t) = 30 - t/2 where
a) Let´s find the function of the population of prairie dogs P(t). For that, let´s integrate the P´(t) function between t = 0 and t and between P = 30 and P
P(t) = ∫P´(t)
P´(t) = dP/dt = 30 - t/2
Separating variables:
dP = (30 - t/2) dt
∫dP = ∫(30 - t/2) dt
P - 30 = 30 · t - t²/4
P(t) = 30 + 30 · t - t²/4
The population of prairie dogs at t = 9 months will be equal to P(9):
P(9) = 30 + 30(9) - (9)²/ 4
P(9) = 280 prairie dogs
The population of prairie dogs after nine months is 280.
b) P(t) = 30 + 30 · t - t²/4 (it was obtained in part a).
